Introduction of Graphs

A graph consists of two sets V and E.

• V is finite and non empty set of vertices
• E is a set of edges connecting two vertices. 

Graphs can be directed or undirected

Directed Graph: If all the edges of a graph have specific direction then this type of graph is called directed graph.For Example: V = {1,2,3,4} and E={(1,2), (2,3), (2,4), (3,1), (4,1), (4,3)}Then Directed Graph G={V,E} represented as

Undirected Graph: If all the edges of the graphs has no specific direction then this type of graph  is known as undirected graph.For Example: V = {1,2,3,4} and E={(1,2), (1,3),(1,4), (2,1),(2,3),(2,4), (3,1),(3,2),(3,4), (4,1),(4,2), (4,3)}Then Undirected Graph G={V,E} represented as

 Representation of Graph

Adjacency Matrix: It is the two dimensional array or matrix representation of graph. In the matrix, the element a(i,j)= 0 if there is no edge between vertex i and j; the element a(i,j)=1 if there is an edge between vertex i and j. It is also known as Static Representation of graph.

For Example : Consider the following Directed Graph.  The Adjacent Matrix is 

        1    2     3     4

1      0    1     0     0

2      0    0     0     1

3      0    0     0     0

4      1    0     1     0




Consider the following Undirected Graph.  The Adjacent Matrix is  

        1    2     3     4

1      0    1     0     1

2      1    0     0     1

3      0    0     0     0

4      1    1     1     0

Adjacency List: It is the list representation of graphs. In this representation, if there are n vertices then there are n linked list for each vertex of graph. It is also known as Dynamic Representation of graph.

Traversing a Graphs

Exploring each vertex of the graph is known as traversing a graph. The order in which we explored the vertices depend upon what kind of data structure is used.  The two common graph traversal method are

1. Breadth First Search (BFS)
2. Depth First Search (DFS)

Spanning Tree 

A tree T is a spanning tree of a graph G={V,E} if every vertex of G belongs to an edge in T and theedges in T form a tree. Details...

Graph Traversal - Breadth First Search

Breadth First Search can be used to find the shortest distance between some starting vertex and remaining vertex of the graph. In this method, all the vertices are stored in a Queue and each vertex of the graph is visited or explored once. The oldest vertex (added first) in the Queue is explored first. To traverse the graph using breadth first search, Adjacent List of a graph is created first.

EXAMPLE

For Example : Consider the following Graph

The Adjacent List is

11 -> 12,18,13

12-> 11,14,15

13-> 11,16,17

14-> 12,18

15-> 12,18

16-> 13,18

17-> 13,18

18-> 14,15,11,16,17

Let us suppose that starting node is 11, The steps to traverse graph using Breadth First Search Is

Step 1: Add 11 to queue

Front: 1                    Queue: 11

Rear:1

Step 2: Delete element at Front  position (11) and add all the adjacent nodes of deletd not that are not visited.

Front: 2                    Queue: X,12, 18, 13

Rear:4

Step 3: Delete element at Front position (12) and add all the adjacent nodes of deletd not that are not visited.

Front: 3                    Queue: X,X,18, 13, 14, 15           

Rear:6

 11 already visited; i.e. in Queue or deleted

Step 4: Delete element at Front position (18) and add all the adjacent nodes of deletd not that are not visited.

Front: 4                    Queue: X,X,X,13, 14, 15, 16,17        

Rear:8

11,14,15 already visited; i.e. in Queue or deleted

Step 5: Delete element at Front position (13) and add all the adjacent nodes of deletd not that are not visited.

Front: 5                    Queue: X,X,X,X, 14, 15, 16,17        

Rear:8

No vertex added becoz 11,16,17 already visited; i.e. in Queue or deleted

Step 6: Delete element at Front position (14) and add all the adjacent nodes of deletd not that are not visited.

Front: 6                    Queue: X,X,X,X, X, 15, 16,17       

Rear:8

No vertex added becoz 12 and 18 already visited; i.e. in Queue or deleted

Step 7: Delete element at Front position (15) and add all the adjacent nodes of deletd not that are not visited.

Front: 7                    Queue: X,X,X,X, X,X,16,17      

Rear:8

No vertex added becoz 12 and 18 already visited; i.e. in Queue or deleted

Step 8: Delete element at Front position (16) and add all the adjacent nodes of deletd not that are not visited.

Front: 8                    Queue: X,X,X,X, X,X,X,17      

Rear:8

No vertex added becoz 13 and 18 already visited; i.e. in Queue or deleted

Step 9: Delete element at Front position (17) and add all the adjacent nodes of deletd not that are not visited.

Front: 8                   Queue: X,X,X,X, X,X,X,X      

Rear:8

No vertex added becoz 13 and 18 already visited; i.e. in Queue or deleted

Queue EMPTY. STOP.

Therefore Breadth First Search is 11,12,18,13,14,15,16,17

ALGORITHM

Step 1: Initialize all the vertices and create adjacent list.

Step 2: Put Starting vertex in Queue.

Step 3: Repeat Step 4 and 5 until Queue is empty.

Step 4: Remove front node from the Queue.

Step 5: Add to the rear of the Queue all the neighbors of the deleted node that are not visited or added in a Queue.

Step 6: Exit

APPLICATIONS

1. Used to test whether graph is connected or not
2. To find the cycles in the graphs
3. To find the path with minimum number of edges between start vertex and current vertex or reporting that no such path exists.

Graph Traversal - Depth First Search

Depth First Search is used to perform traversal of a graph. In this method, all the vertices are stored in a Stack and each vertex of the graph is visited or explored once. The newest vertex (added last) in the Stack is explored first. To traverse the graph, Adjacent List of a graph is created first.

EXAMPLE

For Example : Consider the following Graph

The Adjacent List is

11 -> 12,18,13

12-> 11,14,15

13-> 11,16,17

14-> 12,18

15-> 12,18

16-> 13,18

17-> 13,18

18-> 14,15,11,16,17

Let us suppose that starting node is 11, The steps to traverse graph using Depth First Search is

Step 1: Push 11 onto the Stack

Stack: 11

Step 2:  Pop the top element i.e. last added (11) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack:  12,18,13

Step 3:  Pop the top element i.e. last added (13) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack: 12,18,16,17

11 not added becoz already visited; i.e. onto stack or deleted

Step 4:  Pop the top element i.e. last added (17) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack: 12,18,16

13 and 18 not added becoz already visited; i.e. onto stack or deleted

Step 5:  Pop the top element i.e. last added (16) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack: 12,18

13 and 18 not added becoz already visited; i.e. onto stack or deleted

Step 6:  Pop the top element i.e. last added (18) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack: 12,14,15

11,16 and 17 not added becoz already visited; i.e. onto stack or deleted

 

Step 7:  Pop the top element i.e. last added (15) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack: 12,14

12 and 18  not added becoz already visited; i.e. onto stack or deleted

Step 8:  Pop the top element i.e. last added (14) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack: 12

12 and 18  not added becoz already visited; i.e. onto stack or deleted

 

Step 9:  Pop the top element i.e. last added (12) from the stack and push all adjacent vertex of popped vertex onto the stack that are not visited.

Stack: ----

11, 14 and 15  not added becoz already visited; i.e. onto stack or deleted

Stack Empty. Therefore the Depth First Search is  11,13,17,16,18, 15,14,12

METHOD

Step 1: Intialise all the vertices.

Step 2: Push starting vertex onto the Stack.

Step 3: Repeat step 4 and 5 until Stack is empty.

Step 4: Pop the top vertex from the Stack.

Step 5: Push onto stack all the neighbours of Popped vertex

Step 6: Exit

ALGORITHM

Step 1: currentNode=startNode

Step 2: visited[startNode]=1

Step 3: Repeat

             { for all vertex w adjacent from currentNode

                 {      if (visited[w]==0)

                                { PUSH(w)

                                visited[w]=1

                                }

                 }

           If Q is empty then return

                Else  POP(currentNode)

         }

Graph - Minimum Spanning Tree

DEFINITION 

Let G=(V,E,W) be any weighted graph. Then a spanning tree whose cost is minimum is called minimum spanning tree. The cost of the spanning tree is defined as the sum of the costs of the edges in that tree.

The two method's used are

1. Prim's Algorithm
2. Kruskal's Algorithm

Traversing Binary Tree

DEFINITION

Traversing a Binary Tree means processing / visiting all the nodes of a Binary Tree once in a systematic manner. Starting from the root of a binary tree, there are three main methods that can be performed to perform tree traversal. These methods are

• Inorder Traversal
• Preorder Traversal
• Postorder Traversal

TRAVERSAL METHODS

Inorder Traversal 

To traverse a non-empty binary tree in inorder, the following operations are performed recursively
   1. Traverse the left subtree.
   2. Visit the root.
   3. Traverse the right subtree.

Algorithm

For a given binary tree let the address of root node is given by the pointer R. Let LPTR stores address if left child and RPTR stores address of right child. This algorithm recursively traverse the binary tree in  inorder

INORDER(T)
Step 1: Check for empty Tree
             IF R=NULL THEN
                  PRINT ("EMPTY TREE")
                  EXIT
            ENDIF

Step 2: Traverse Left subtree in inorder
            IF LPTR(R) < > NULL THEN
                  CALL INORDER(LPTR(R))
            ENDIF

Step 3: Process the Root
            PRINT (INFO(R))

Step 4: Traverse Right subtree in inorder
             IF RPTR(R) < > NULL THEN
                  CALL INORDER(RPTR(R))
             ENDIF

Step 5: Exit

Preorder Traversal

To traverse a non-empty binary tree in preorder, the following operations are performed recursively 
   1. Visit the root.
   2. Traverse the left subtree.
   3. Traverse the right subtree.

Algorithm

For a given binary tree let the address of root node is given by the pointer R. Let LPTR stores address if left child and RPTR stores address of right child. This algorithm recursively traverse the binary tree in  pre order

PREORDER(T)
Step 1: Check for empty Tree
             IF R< > NULL THEN
                  PRINT INFO(R)
             ELSE
                  PRINT ("EMPTY TREE")
                  EXIT
            ENDIF

Step 2: Traverse Left subtree in preorder

            IF LPTR(R) < > NULL THEN
                  CALL PREORDER(LPTR(R))
            ENDIF

Step 3: Traverse Right subtree in preorder
             IF RPTR(R) < > NULL THEN
                  CALL PREORDER(RPTR(R))
             ENDIF

Step 4: Exit

Postorder Traversal

To traverse a non-empty binary tree in postorder, the following operations are performed recursively 
   1. Traverse the left subtree.
   2. Traverse the right subtree.
   3. Visit the root.

Algorithm

For a given binary tree let the address of root node is given by the pointer R. Let LPTR stores address if left child and RPTR stores address of right child. This algorithm recursively traverse the binary tree in  postorder

POSTORDER(T)
Step 1: Check for empty Tree
             IF R=NULL THEN
                  PRINT ("EMPTY TREE")
                  EXIT
            ENDIF

Step 2: Traverse Left subtree in postorder

            IF LPTR(R) < > NULL THEN
                  CALL POSTORDER(LPTR(R))
            ENDIF

Step 3: Traverse Right subtree in postorder
             IF RPTR(R) < > NULL THEN
                  CALL POSTORDER(RPTR(R))
             ENDIF


Step 4: Process the Root
            PRINT (INFO(R))


Step 5: Exit